What is the significance of the play’s title?

Petruchio "tames" the "shrews" behavior. The shrew is Kate, and he must tame her because in those days...... men wore the pants in the family. Kate's character definitely pushed the submission expected of a woman.

Petruchio to Kate:

Why came I hither but to that intent?

Think you a little din can daunt mine ears?

Have I not in my time heard lions roar?

Have I not heard the sea puff'd up with winds

Rage like an angry boar chafed with sweat?

Have I not heard great ordnance in the field,

And heaven's artillery thunder in the skies?

Have I not in a pitched battle heard

Loud 'larums, neighing steeds, and trumpets' clang?

And do you tell me of a woman's tongue,

That gives not half so great a blow to hear

As will a chestnut in a farmer's fire?

Tush, tush! fear boys with bugs.